The Morse–Kelley set theory () is an axiomatic approach to set theory which has both classes and sets. Whereas NBG (which also has both classes and sets) is conservative? over ZFC, Morse–Kelley is not a conservative extension of . The principal difference from is that allows arbitrary formulas appearing in the class comprehension axiom schema (in particular, formulae with quantifiers ranging over classes themselves).
The approach is explained in the appendix to John Kelley’s 1955 book General Topology.